Composition Games for Distributed Systems: the EU Grant games

Composition Games for Distrib uted Systems: the EU Grant games ∗ Shay Kutten Ron Lavi Amitabh T rehan Faculty of Industrial Engineering and Management T echnion, Haifa, Israel Nov ember 23, 2021 Abstract W e analyze ways by which people decompose into groups in distributed systems. W e are interested in systems in which an agent can increase its utility by connecting to other agents, but must also pay a cost that increases with the size of the system. The right balance is achiev ed by the right size group of agents. W e formulate and analyze three intuitiv e and realistic games and show how simple changes in the protocol can drastically improve the price of anarchy of these games. In particular, we identify two important properties for a low price of anarchy: agreement in joining the system, and the possibility of appealing a rejection from a system. W e show that the latter property is especially important if there are some pre-existing constraints regarding who may collaborate (or communicate) with whom. 1 Intr oduction It is likely that agents will form groups when they gain from cooperation. In peer- to-peer (P2P) systems, agents team up to share content; in multi-agent systems, agents team up to complete a task. This moti vation to cooperate is a “force” that pushes to wards grouping, and has been addressed in numerous papers (some are surve yed below). Here, we are interested in the combination of this force with a second force that breaks lar ge groups into smaller ones. A group that is too large may incur costs that are too high, such as ov erheads, free loaders, e xposure to out- side threats (e.g. lawsuits ov er intellectual properties), etc. This may decrease the ∗ This work was carried out in and supported by the T echnion-Microsoft Electronic-Commerce Research Center . 1 v alue agents get from a large group and motiv ate them to break it. Also, consider biological ecosystems: A species with too fe w members will die out due to com- petition and under exploitation of resources; if the species has too many members and the resources are fixed or slow to rene w , the species may die out due to ov er- exploitation. This suggests that for many systems, the sweet spot lies somewhere in the middle. Consider the example of FP7, the current, 7th Frame work Programme for sup- porting research in Europe. One of its main goals is to form large “networks of excellence”. Of course, the commission is not satisfied with size alone, it is in- terested in the combination of size and quality . How can the commission obtain a “network” satisfying both of these criteria? Clearly , the commission cannot simply choose some set of researchers to its liking, telling them that they now form a re- search network, and must cooperate to obtain good research results. Usually , it is hard for the commission to find a researcher’ s value unless the researcher exhibits this v alue first, by submitting a proposal. In addition, the v alues of researchers ma- terialize when they work with researchers with whom they find common grounds and like to cooperate. This scenario is similar to other cases where agents in a distributed system decompose into smaller groups. For example, the factors mentioned above seem to be present in a P2P system as well. First, people form such systems voluntarily . Second, there may not be a grant, but the members do realize some benefit from pooling their resources (music, movies, etc.) together . Third, we already noted that a motiv ation is often present, for these people to form a more exclusi ve system, rather than to hav e a very lar ge one. T o capture these and other realistic settings, we study a model in which a de- signer seeks to choose a connected subset of nodes in an underlying network, where each node has a quality parameter . A group of nodes can perform the task only if the sum of their qualities passes a certain threshold, T (we term such a group “eli- gible”). Having passed this threshold, the success of a group positi vely depends on its av erage quality . The designer (granting agency) wishes to award a grant of M Euros to the best such group. The winning group should also decide ho w to partition the grant money among its members. While it may be possible for the agents to bargain on this issue, in practice many times this is not the case. Probably due to strong social norms, researchers simply split the money ev enly . This happens in many other settings as well. In P2P systems, the prize is the shared content, to which everyone has equal access. In this paper , we study group composition only under the assumption that the grant is split ev enly . There is no doubt that other ways to split the grant are possible, but we defer the study of this issue to future research. 2 In this work we introduce three natural protocols/games for deciding the com- position of groups, and study their price of anar c hy (PO A): [13] the ratio between (a) the optimal (maximal) av erage quality of an eligible set of researchers, and (b) the lowest average quality of a winner set that can be formed in an equilibrium (here we use both the notions of a Nash equilibrium and of a strong equilibrium). PO A is a measure of the degradation of the efficienc y of a system due to self- ish behavior of agents; higher ratio corresponds to higher loss and poorer quality worst case equilibria. W e remark that the use of non-cooperativ e game theory fits our goals better , compared to using cooperativ e game theory . The latter theory is mainly concerned with the correct and most efficient way to distribute payof fs among members of a winning coalition, hiding the workings of how the winning coalition is formed. Howe ver , our interests are dif ferent and focus on the question: what is the composition of the winning gr oup, and what is its quality ? Moreover , in our model the answer to “which group wins” depends not only on the group, but also on the competing groups. The first protocol, the gold rush game, is a naiv e composition method often used in some le gacy systems (e.g. mailing lists): joining a group is done by simply declaring (unilaterally) the will to do so. Granting agencies do not usually use such a method. Perhaps the reason is that, as we show , its price of anarchy is very high, bounded only by the size of the whole society . The analysis of this game is trivial, but forms a basis for comparison, and is a good w armup. Usually , granting agencies require some stronger condition for researchers to join a group: at the least, that all group members agree on its composition (the list of participants). If this method is used and the underlying collaboration network is a clique (everyone knows e veryone else), the str ong price of anarc hy improv es drastically , to be at most 2. While this improv ement is impressiv e, we also show that when the underlying collaboration network is arbitrary (in particular , not a clique), the strong price of anarchy of this method can grow up to 3, which means that only a third of the optimal av erage quality may be realized. While the source of the high price of anarchy of the first method was the “too easy” joining, the source of the (still rather) high price under the second method is just the opposite – the dif ficulty of joining. That is, we show examples where some winning set finds it beneficial to disallo w the joining of some high value re- searchers. W e introduce a third protocol as a simple alleviation of the pre vious problems: allowing those rejected high v alue researchers to “appeal” the rejection. (Interestingly , a granting agenc y named MA GNET , of the Israeli ministry of indus- try , uses a similar method for the joining of companies to a consortium). W e show that the strong price of anarchy of this third method is lo wer: at most 2 on arbitrary networks, and even approaches 1 (the optimum) for large sets, if the collaboration network is a complete graph. The analysis of this third method constitutes the main 3 technical contrib ution of this paper . It sho ws how the topology of the collaboration network af fects quality , and what is the importance of an appeals process. 2 Related Literature Probably , the most related paper to ours is [5]. They , too, use non-cooperativ e game theory . Moreover , in their model, too, agents hav e values, and the sum of the values of agents in a winning coalition must exceed a fixed threshold. Howe ver , that model seems to focus on points that do not capture our moti v ating scenarios. First, we, intentionally , focus on the competition part, and assume that the payoff to the winning group is fixed. In contrast, there, the values of the players in the winning coalition are also the total payoff of that coalition. This assumption drastically af fects the nature of the competition. Second, we, intentionally , focus on situations where the social norm or physical reality dictates the equal sharing of the grant money; our aim is to characterize the coalitions (and qualities) that will result from the giv en division of the total gain. In contrast, they aim to analyze the bargaining process through which the division of the total gain from the cooperation will be determined. Finally , our price of anarchy e v aluates the values of the winning group. In contrast, the price of democracy they analyze becomes meaningless in our case where there are no costs (costs are an orthogonal parameter introduced there). [12] studies cooperativ e games over graphs, where only connected coalitions S are able to extract their value v ( S ) . This is also one of our assumptions. They sho w that the unique fair way to divide the value of the grand coalition is by the Shapley v alue. The current paper does not deal with di viding the v alue. Moreov er , here, a group may or may not win, as a function of the the actions of the players in competing groups. This does not seem to be captured well by cooperati ve game theory . While most works in classic cooperativ e g ame theory are only remotely related to our specific model, a series of papers on the stability of coalition structures, see for example [6], [4], and references therein, is quite relev ant to our work. They study a setting where society splits into dif ferent coalitions, and characterizes cases where such partitions are stable. (This seems more related than the famous stable marriage problem [8].) Unlike the current paper , they assume that all formed coali- tions win a prize (otherwise, clearly , no stable coalition structure will emerge). An- other difference is that these papers do not focus on specific protocols for forming coalitions; their main interest is in characterizing when such stable structures exist. [9] models the creation of coalitions in a game that bears similarities to our initial example game (the gold-rush). Howe v er , they do not aim to analyze the quality of the resulting coalition. 4 Our paper is also related to the literature on network creation games, starting with [7]. These papers study games in which nodes decide how to form links in order to create a connected network, and the price of anarchy is analyzed under v arious assumptions. In their models, the society becomes connected . In contrast, in our paper , the society splits to giv e birth to a strict subset , and the question is whether the quality of the formed group is far from optimal because of various strategic issues. The literature on price of anarchy is rich, see [13] for a surve y . 3 T w o Protocols with Opposing P olicies A granting agency wishes to award a prize of M Euros to a subset of a society of n researchers. Each researcher i has a v alue v i that represents her overall quality . A subset of researchers is eligible if (i) their sum of values is at least some giv en threshold T , and (ii) they form a connected component of the underlying Collab- oration Network (CN). For simplicity , in most of this section, we assume that the underlying collaboration network is the complete graph, but remove this assump- tion in subsequent sections. The granting agency aims to aw ard the prize to a set of researchers (“consortium”) with maximal av erage quality among all eligible con- sortia. T o exclude some trivial cases, we assume throughout that v i < T for ev ery researcher i , i.e. the researcher with the maximal v alue cannot take the prize on her o wn. W e also assume, without loss of generality , that the sum of all v alues is larger than T . While the agency does not know the values of the researchers, we assume that it can verify the values when a set of researchers submits evidence of their v alue (this is the “grant proposal”). The agency constructs a protocol, by which researchers form candidate consortia, and the best formed consortium wins and re- cei ves the prize. T wo natural protocols gi ve some intuition for possible causes of a high price of anarchy . The Gold-Rush Game. This protocol, as well as its analysis, are trivial. How- e ver , they serve as a basis for comparison, as well as a “warm up example”. Each researcher submits a separate proposal, reporting (along with a proof of the re- searcher’ s value) some label, the “consortium name”. The labels are taken from some finite set of labels L . Researchers who report the same label are understood to belong to the same consortium. The agency awards the prize to an eligible con- sortium with the maximal av erage value (in case of ties, any arbitrary (possibly randomized) tie-breaking rule can be used). Each researcher in the winning con- sortium recei ves an equal share of the prize. In terms of game theory , the strate gy of each researcher i in this game is the label ` i she chooses. The utility u i ( ` 1 , ..., ` n ) of i is 0 if “her” consortium loses, and M y if her consortium wins, where y is the size of the winning con- 5 sortium. A tuple of strategies ` 1 , ..., ` n is a Nash equilibrium if u i ( ` 1 , ..., ` n ) ≥ u i ( ` 1 , ..., ` i − 1 , ` 0 i , ` i +1 , ..., ` n ) for every i = 1 , ..., n and every ` 0 i ∈ L . In other words, in a Nash equilibrium ` 1 , ..., ` n , the utility of each researcher i is maximized by declaring ` i , giv en that the other researchers declare ` − i = ` 1 , ..., ` i − 1 , ` i +1 , ..., ` n . It has become standard in the algorithmic game theory literature to measure the quality of a game/protocol by its price of anar chy (PO A) [10]. In our case, this is the optimal (largest possible) average value divided by the av erage value of the winning consortium in the worst Nash equilibrium. (This reflects a worst-case point of vie w). Unfortunately , the price of anarchy of the gold-rush game is very high. T o sho w this, it suffices to study the case of distinct v alues (i.e. no two v alues are equal) and a complete CN. T o analyze the price of anarchy , the next lemma characterizes all Nash equilibria of this game. (Most proofs in this section are deferred to the journal version of this paper (arxi v v ersion at [11])). Lemma 1. Assume that values ar e distinct and the CN is the complete gr aph. Then, in every Nash equilibrium of the gold-rush game either no consortium forms, or all r esear c hers declare the same label, hence all r esear cher s win. This immediately implies an unbounded price of anarchy: Theorem 1. The price of anar chy of the gold-rush game is (arbitrarily close to) n/ 2 . There is another , more conceptual problem with the gold-rush game. In real- ity , researchers (as well as P2P users) may know each other , and can coordinate a joint deviation from the presumed equilibrium strategy , e.g. the top-value re- searchers may coordinate to belong to an exclusi v e consortium. The notion of a Nash equilibrium does not allow such coordinated deviations, and is therefore conceptually weak for our case. A better notion is a str ong (Nash) equilibrium , which requires that no subset of the players can jointly de viate and increase each of their utilities [3]. Formally , a tuple of strategies ` 1 , ..., ` n is a str ong equi- librium if for any ` 0 1 , ..., ` 0 n ∈ L there exists a player i such that ` 0 i 6 = ` i and u i ( ` 1 , ..., ` n ) ≥ u i ( ` 0 1 , ..., ` 0 n ) . Howe ver , consider the situation where there exists at least one eligible group that is a strict subset of the society . One of these groups has maximal av erage value and should be a winner . Informally , all researchers not in this group will like to switch their labels to the winners b ut the winners will like to defect as a group to a different label by themselves. This leads to the following lemma: Lemma 2. If there exists an eligible gr oup which is a strict subset of the society , ther e does not e xist even a single str ong equilibrium in the gold-rush game. 6 Consensual consortium composition (CCC). Intuitiv ely , the bad price of anar- chy of the gold-rush game resulted from the fact that it was “too easy” for anybody to join a consortium of her liking. The following Consensual Consortium Com- position (CCC) game is a first attempt to fix the problems of the previous nai ve design. In this game, each player submits a “proposal:” her value and a list of the researchers in her consortium. An eligible consortium of researchers X then satisfies (1; consistency) each researcher in X submitted X as her consortium, (2; threshold) P i ∈ X v i ≥ T , and (3; connectivity) the consortium is connected in the underlying CN. The winning consortium is an eligible consortium with maximal av erage v alue. If sev eral such consortia exist, the winning one has minimal size. As discussed abov e, Nash equilibrium is not really appropriate in our context. In fact, for the CCC game, a Nash equilibrium is meaningless. The reader can verify that in this game, any partition of the players into consortia will constitute a Nash equilibrium. W e focus on the stronger and more appropriate notion of a strong equilibrium. Analogous to PoA, strong price of anarchy (SPOA) [1] in our case is the ratio of the largest average value to the average value of the winning consortium in the worst strong equilibrium. Theorem 2. Assume that CN is a clique. F ix an arbitrary tuple of r esear cher values, and suppose that a minimal eligible consortium with the highest avera ge value has size k . Then, the str ong price of anar chy of the CCC game is (arbitrarily close to) 1 + 1 k − 1 . In particular , the SPO A of the CCC game is at most 2. Pr oof. Assume the CN is the complete graph. Fix a tuple of values v 1 , ..., v n . Let O P T be a minimal eligible consortium with highest average value, and denote | O P T | = k . Let i ∈ O P T be a player with a lowest value among all players in O P T . Then, P j ∈ O P T \{ i } v j < T : if all values in O P T are equal, this follows from the minimality of O P T . If not all values in O P T are equal the inequality follo ws since otherwise O P T \ { i } is an eligible consortium with a higher average v alue than O P T . The a verage value of OP T \ { i } is, therefore, at most T k − 1 . Thus, the av erage v alue of O P T is also at most T k − 1 . No w , let W be the winner consortium in some strong equilibrium. If | W | = l < k then there exists an eligible consortium with l < k researchers. Thus, the l players with l highest values form an eligible consortium with average value not smaller than that of O P T , a contradiction. If | W | > k then W cannot be a strong equilibrium. This is because players in OP T can de viate, form a consortium, win, and increase their utility (since the size of the winning subset strictly decreases). Thus, | W | = k . Since the sum of values of players in W is at least T , the average v alue of players in W is at least T k . 7 By the previous conclusions, the ratio of the average v alue of players in O P T to the av erage value of players in W is at most k k − 1 . This prov es that the SPO A of the CCC game is at most 1 + 1 k − 1 . T o pro ve a matching lo wer bound, an example (for e v ery gi ven k ) suf fices. For this purpose, consider the following tuple of values, for any small enough  > 0 . There are k researchers 1 , ..., k , all with the same value T k − 1 −  , and researcher k + 1 with value k  . In this case, the eligible consortium with the highest average v alue (of T k − 1 −  ) is { 1 , ..., k } . Having researchers { 1 , ..., k − 1 , k + 1 } form one consortium, excluding researcher k , is a strong equilibrium – one can verify that no subset can de viate and strictly increase each of their utilities. Thus, the SPOA in this case approaches k k − 1 as  approaches 0 . When the CN is not a complete graph, this theorem is not necessarily true. Figure 1 shows an example of a non-clique CN and player v alues such that the SPO A is 3 −  , where  is an arbitrarily small constant. W e conjecture that 3 is the correct bound. W e remark that for the CCC game a strong equilibria always exists, see Lemma 4. The figure demonstrates another interesting phenomenon: the optimal consortium is not necessarily a strong equilibrium. Here, the central node will pre vent the formation of the optimal group. T − 2 T − 2 ε ε ε ε T − 2 T − 2 ε ε k Figure 1: The SPO A for a CCC game on a CN can be arbitrarily close to 3. Here, the nodes are labeled with their values: T is the threshold,  is an arbitrarily small v alue. The worst equilibria is shown in the box: the other nodes and the central  node form the optimal group. 4 Main Result: MA GNET CCC Game for arbitrary Col- laboration Networks As shown above, the quality of the winning group of the CCC game may be only one third of the optimal quality . W e show how to improv e the SPO A to 2 for any collaboration network; for the complete graph this it will become close to 1. F or this purpose, we introduce an extension of the CCC game which proceeds over multiple rounds (the name is inspired by a policy of the MA GNET Israeli granting 8 agency which uses a similar policy). The MA GNET CCC Game is defined as follo ws: • In round 1, ex ecute the CCC game. Let the winning consortium be W 1 . In each round r > 1 the winning consortium W r is an expansion of W r − 1 . • In round r > 1 , each researcher not in W r − 1 can “submit an appeal” – a proposal consisting (as in CCC) of e vidence of v alue and a list of researchers in her consortium. The winning consortium W r in round r is the union of W r − 1 and all appealing consortia X that satisfy (1; connectivity) X ∪ W r − 1 is a connected component in CN , (2; consistency), each researcher in X submitted X as her consortium, and (3; Improvement) av g (X ∪ W r ) > av g ( W r − 1 ) . • The game ends if W r = W r − 1 (no justified appeals). W e next analyze the SPO A of this game, first for any arbitrary CN , then for specific graph structures. 4.1 Analysis of SPO A for arbitrary CN This section shows the main technical result of the paper: The MA GNET CCC Game has SPOA that is equal to exactly 2, regardless of the topology of the CN. T o prove this, we first identify some properties that any winning consortium in the MA GNET CCC Game must have. Throughout, we denote by SO W (Social Optimum W inner) a minimal eligible consortium among all eligible consortia with maximal av erage v alue. Lemma 3. Let Z be the winning consortium in some str ong equilibrium outcome of some arbitrary instance of the MA GNET CCC Game. Then, 1. Z ∩ S O W 6 = ∅ 2. | Z | ≤ | S O W | 3. av g ( Z ) ≥ av g ( S OW \ Z ) Pr oof. 1. Z ∩ S O W 6 = ∅ : If Z is not an SO W , Z can win only if it can pre vent the formation of S O W . This can happen only if Z has some member(s) of S O W i.e. Z ∩ S O W 6 = ∅ . 2. | Z | ≤ | S O W | : If | Z | > | S O W | , the players in Z ∩ S O W can impro ve their utility by forming the smaller consortium S O W in the first round, which is a sure winner (having the highest a v erage). Thus, | Z | ≤ | S O W | . 9 3. av g ( Z ) ≥ av g ( S O W \ Z ) : If av g ( Z ) < av g ( S O W \ Z ) , the players in S O W \ Z can become winners which is a contradiction. They can appeal to- gether after the currently last round. Since both S O W and Z are connected, so is S O W ∪ Z . Since av g ( Z ) < av g ( S O W \ Z ) , S O W \ Z can be added as winners. Lemma 4. The MA GNET CCC Game always has a Str ong Equilibrim (S.E.) Pr oof. If S O W is not a S.E., there is a winning consortium Z having non-empty intersection with S O W and of size strictly smaller than S O W . If Z is not a S.E., some nodes of Z can deviate. I.e., these nodes (maybe with other nodes) can form a consortium Z 0 having non-empty intersection with S O W and of size strictly smaller than Z . Since this process is finite, we must reach a subset Z 00 that is a S.E. Theorem 3. The SPO A of MA GNET CCC Game ≤ 2 . Pr oof. Denote | S O W | = k . By definition, S P O A = av g ( S O W ) av g ( Z ) = sum ( S O W ∩ Z ) k + sum ( S O W \ Z ) k av g ( Z ) (1) W e prov e two properties: 1. sum ( S O W ∩ Z ) /k ≤ av g ( Z ) : obviously sum ( S O W ∩ Z ) ≤ sum ( Z ) . By Lemma 3, | Z | ≤ k . Thus, sum ( S O W ∩ Z ) k ≤ sum ( Z ) k ≤ sum ( Z ) | Z | = av g ( Z ) 2. sum ( S O W \ Z ) /k ≤ av g ( Z ) : By Lemma 3, av g ( Z ) ≥ av g ( S O W \ Z ) . Thus, sum ( S O W \ Z ) k ≤ sum ( S O W \ Z ) | S O W \ Z | = av g ( S O W \ Z ) ≤ av g ( Z ) Plugging into equation 1, we get: S P O A = sum ( S O W ∩ Z ) /k + sum ( S O W \ Z ) /k av g ( Z ) ≤ av g ( Z ) + av g ( Z ) av g ( Z ) = 2 10 ε T− ε n−2 0 0 0 0 n−2 T(n−1)/n Figure 2: For the MA GNET CCC Game a Collaboration Network with a price of anarchy arbitrarily close to 2. Figure 2 shows an example with S P O A = 2 −  , where  is an arbitrarily small constant. The S O W consists of the two players with values T ( n − 1) n and T −  , coupled with their intermediary players that hav e zero values. The worst strong equilibrium here is the consortium containing the two players with values  and T −  , coupled with their intermediary players that have zero v alues. This gi ves S P O A = 2 n − 1 n −  T , which can be made arbitrarily close to 2. 4.2 Dependency on graph parameters The pre vious analysis showed a SPOA of 2 for arbitrary graph structures. W e show ho w the SPO A may depend on the structure of the graph. In particular , we consider two extreme special cases: the complete graph on one hand, and the line network on the other hand. These two cases result from differences in the diameter and the connecti vity . The Complete Graph. In the CCC game, this case has SPO A 1 + 1 k − 1 ( k = | S O W | ). The multiple round version improves this bound to be 1 + 1 k . While the improv ement is small for large S O W ’ s, for small S O W ’ s it is quite significant, e.g. the SPO A can decrease from 2 (with one round) to 1.5 (with multiple rounds). Theorem 4. The SPO A of the MA GNET CCC Game over a complete C G is exactly 1 + 1 k , wher e k = | S O W | . Pr oof. Assume without loss of generality that v 1 ≥ v 2 ≥ · · · ≥ v n . The proof relies on the following two observations (full proof deferred to the journal version): Observation 1. F or the case of the complete graph, every S O W consortium has size k , wher e k is suc h that P k − 1 i =1 v i < T and P k i =1 v i ≥ T . Observation 2. F or the case of the complete graph, the size of any winner con- sortium in a str ong equilibrium outcome must be equal to the size of the SO W consortium. 11 Lemma 5. Ther e e xists an instance of the MAGNET CCC Game over a complete CN for which the str ong price of anarc hy is arbitrarily close to 1 + 1 k , wher e k = | S OW | . Pr oof. Fix k , and consider the following tuple of values, for any  > 0 : There are k − 2 researchers 1 , . . . , k − 2 with the same v alue T k − 1 . Researchers k − 1 , k , k + 1 hav e values T k − 1 −  , T k , and  respecti vely . In this case, the eligible consortium with the highest average is 1 , . . . , k having av erage T + T /k −  k . The worst strong equilibrium has the winner as the consortium 1 , . . . , k − 1 , k + 1 . This has a verage T k , hence the strong price of anarchy approaches 1 + 1 k as  approaches 0. The Line Network. This is the other extreme. Here, the SPO A gr ows and ap- proaches 2 as the size of the SO W increases (Theorem 5). In contrast, in a complete graph (as shown above), the SPO A shrinks and approaches 1 as the size of the SO W increases. Intuitiv ely , it seems that this 100% increase (from the optimum) in the case of a line results from the growth of the diameter when k grows. On the other hand, the disappearance of the price of anarchy (conv ergence to 1) in the case of a complete graph seems to result from the increase in connecti vity . Theorem 5. In the MA GNET CCC Game o ver a line CN , the SPO A is (arbitrarily close to) 1 + k − 1 k ( k = | S OW | ). Pr oof. Let W be the winner in a worst strong equilibrium, and let k 0 = | S O W \ W | . By Lemma 3, k 0 ≤ k − 1 (since S O W ∩ W is not empty), and av g ( S O W \ W ) ≤ av g ( W ) . Hence, sum ( S O W \ W ) ≤ k 0 · avg ( W ) . By the same lemma, | W | ≤ k , and k · av g ( W ) ≥ sum ( W ) > sum ( S O W ∩ W ) . All these imply: S P O A = av g ( S O W ) av g ( W ) = sum ( S O W \ W ) k + sum ( S O W ∩ W ) k av g ( W ) ≤ k 0 · av g ( W ) k + sum ( S O W ∩ W ) k av g ( W ) = k 0 k + sum ( S O W ∩ W ) k · av g ( W ) ≤ k − 1 k + 1 . The other direction is sho wn via the example in Figure 2, using | S OW | = n (specifically , the SPO A there is 2 n − 1 n −  T = 1 + k − 1 k −  T for an y arbitrarily small  > 0 ). 12 4.3 Additional Issues W e briefly discuss the follo wing two important issues: 4.3.1 Using a notion of strong subgame perfect equilibrium. Since the MAGNET CCC Gameis an extensiv e-form game, one may wonder whether it is more appropriate to use a notion of Strong Subgame Perfect Equilibria (SSPE) instead of Strong Equilibria. W e note that such a change will not change our price of anarchy analysis. First, clearly , every SSPE is also a SE. Thus, using SSPE in- stead of SE can only decrease the price of anarchy . Second, we note that if W is the winner consortium in some strong equilibrium, then there is also a Strong Sub- game Perfect Equilibrium of the MA GNET CCC Game in which W is the winner consortium. This is the tuple of strategies where the researchers in W submit a proposal together already in the first step of the game. In this there will be only one step, and the two notions become the same. Therefore, the price of anarchy is the same, regardless of which notion we use. 4.3.2 Strong Price of Stability Another useful concept analogous to the price of anarchy is the price of stability (POS) [2]. Analogous to SPO A, one can define Strong Price of Stability (SPOS) as the ratio of the optimum to a best strong equilibrium. In our game, this is the ratio of the optimal (largest possible) av erage value to the av erage value of the winning consortium in the best strong equilibrium. W e do not analyze SPOS in detail in this paper , but we wish to note via an example described in Figure 3 that in MA GNET CCC Game, the SPOS can be greater than 1. T/4 T− ε 0 ε ε T− 0 0 0 Figure 3: The Strong Price of Stability for the MA GNET CCC Game can be greater than 1. The figure shows the best Strong Equilibria (green, solid lines) that yields a SPOS of 6/5. For comparison, the worst Strong Equilibria is shown in red, dashed lines sho wing a SPO A of 3/2. 13 5 Conclusions and Future W ork This paper looks at the process of agents teaming up to construct distributed sys- tems. Our setting addresses a specific scenario where one driving force/ incentive limits the size of the consortium, but another increases it. W e made some simple assumptions. W e assumed that the value of a researcher is independent of the mem- bers of its consortium. W e also assumed that the Euro amount of the grant is fix ed. What if these assumptions did not hold? What if the grant were some function of the set size? There are many other interesting directions to explore. W e could have more sophisticated utility functions or game designer goals, or we could study “natural” games (i.e. not design mechanisms but look at existing systems). W e can study more in volv ed environments; such as an e volving dynamic en vironment where new researchers are born and old ones retire. What about composition of multiple sys- tems? Could multiple consortiums form simultaneously or in reaction to other formations? W ould there be a domino effect? Is there a relation between the topol- ogy of collaboration networks and consortium composition? Our work indicates there may be influence of both connectivity and diameter on the SPO A. How does the choice of a threshold (which influences the consortium size) influence SPO A? Can we propose mechanisms that further improv e SPO A? The MAGNET game is a multi-round game. There are known results transforming multi-round games to single round but these in volv e various penalties and assumptions. Can we pro- vide a more efficient reduction in our context? W e hav e assumed the players to be fully rational in their decision making; it will be interesting to study such games in context of bounded rationality and also with players ha ving limited information of their neighborhood as in a distributed netw ork setup. Finally , we would like to abstractly define, e ventually , the class of distributed systems formation games making it easier to understand the v arious trade-of fs and parameters. 6 Acknowledgements W e would like to thank Dahlia Malkhi, Ittai Abraham and Moshe T ennenholtz for their support, and Rann Smorodinsky , Reuven Bar Y ehuda, Liron Y edidsion, Oren Ben-Zwi and the anonymous referees for their comments and discussions. 14 Refer ences [1] N. Andelman, M. Feldman, and Y . Mansour . Strong price of anarchy. 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